Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials.
Let \(W=(W,S)\) be a Coxeter group with \(S\) its distinguished generator set. Denote by \(\leqslant\) the Bruhat ordering on \(W\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined polynomials \(P_{x,y}\in\mathbb{Z}[q]\) for each \(x\leqslant y\) in \(W\) [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. These polynomials play an important role in representation theory. \textit{M. Dyer} conjectured [Compos. Math. 78, No. 2, 185-191 (1991; Zbl 0784.20019)] that \(P_{x,y}\) depends only on the isomorphism type of the poset \([x,y]=\{z\in W\mid x\leqslant z\leqslant y\}\) for the Bruhat ordering. The main result of the present paper is to give an affirmative answer to the conjecture for a class of groups, which can be stated as follows. Let \(W,W'\) be two Coxeter groups. Let \(w\in W\) and \(w'\in W'\) be such that the posets \([e,w]\) and \([e',w']\) are isomorphic for the Bruhat orderings on \(W,W'\), where \(e,e'\) are the identity elements of \(W,W'\), respectively. Then any poset isomorphism \(\phi\colon[e,w]\to[e',w']\) preserves Kazhdan-Lusztig polynomials, (in the sense that \(P_{\phi(x),\phi(y)}=P_{x,y}\) for any \(x\leqslant y\) in \([e,w]\)) in the case where one of the two groups \(W,W'\) has the property that the Coxeter graph of each of its irreducible constituents is either a tree or affine of type \(\widetilde A_n\). In particular, the result holds for all finite or affine Coxeter groups. Note that the above result was also obtained by \textit{F. Brenti} [The intersection cohomology of Schubert varieties is a combinatorial invariant, preprint (2002)] in the case where \(W,W'\) are both of type \(A_n\), as a corollary of a purely combinatorial construction of the \(P_{x,y}\).
scientific article; zbMATH DE number 2022086
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